Ki Song
Posts:
379
Registered:
9/19/09


Re: Vectors summing to zero
Posted:
Sep 2, 2010 6:12 PM


On Sep 2, 7:09 am, "Tim Golden BandTech.com" <tttppp...@yahoo.com> wrote: > > As I go over it again I will stand by the 2 pi claim. If you have a > falsification then please present it. I don't necessarily believe all > of mainstream mathematics so I find it acceptable that you do not > believe mine. Simply stated: the sum of the angles between three > coplanar vectors will always be 2 pi, in any dimension. The converse > holds: If the sum of the angles of three vectors is 2 pi, then the > vectors are coplanar. This is not a proof, but is an observation. I'm > open to falsification, but doubt that you will find one. > >  Tim
I don't really understand what you mean by "sum of the angles between three coplanar vectors will always be 2 pi." I'm assuming you meant dimension greater than or equal to 2 when you said "in any dimension." This statement is trivially false if you just meant "three vectors that that lie in a 2 dimensional subspace."(Take any vector X. Then, take 2X and 3X.) Even if you meant "three vectors that span a 2 dimensional subspace," it is trivially false. (Take two linearly independent vectors, X,Y. Choose X+Y as the third vector. Angle between X&Y, X&(X+Y), and Y&(X+Y) will not add up to 2Pi.)
So obviously, you mean something else. Exactly what do YOU mean by three coplanar vectors?
The converse of the statement is also trivially false. Look at a tetrahedron constructed from 4 equilateral triangles. Imagine that one of the vertices is the origin, and imagine that the three edges are vectors in R^3. What are the angles between the vectors? What is the sum of the angles of these vectors? Are these vectors "coplanar"?
I agree with Gerry that perhaps it would be wise for you to learn a bit of conventional mathematics before rejecting it. If you know some algebra, you'd know that your polysign numbers is just a example of a quotient ring.

